Given that −6i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4−2x3+x2−72x−1260

Question

Given that  is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

f(x)=x4−2x3+x2−72x−1260

Answer 1

Then +6i is a factor...

so (x-6i)(x+6i) is a factor...

Then it divides evenly by x^2 + 36

long division says:

x^2 -2x - 35

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x^2 + 0x + 36 |

x^4 - 2x^3 + x^2 - 72x - 1260

x^4 + 0x^3 + 36x^2

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-2x^3 - 35x^2 - 72x

-2x^3 + 0x^2 - 72x

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-35x^2 - 1260

-35x^2 -1260

Yes, it divides evenly by x^2 + 36

the quotient is x^2 - 2x - 35 = ( x - 7 )( x + 5 )

the factors are (x + 6i)( x - 6i) (x - 7) (x + 5)

the solutions are 7,-5, 6i, -6i

Answer 2